76.15.16 problem 17

Internal problem ID [17586]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 17
Date solved : Monday, March 31, 2025 at 04:18:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=2 t \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)+diff(y(t),t)-2*y(t) = 2*t; 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = {\mathrm e}^{t}-\frac {{\mathrm e}^{-2 t}}{2}-t -\frac {1}{2} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 24

ode=D[y[t],{t,2}]+D[y[t],t]-2*y[t]==2*t; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -t-\frac {e^{-2 t}}{2}+e^t-\frac {1}{2} \]

✓ Sympy. Time used: 0.172 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t - 2*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - t + e^{t} - \frac {1}{2} - \frac {e^{- 2 t}}{2} \]